UV Mapping Rounded Objects & Baking
In my example I marked all the red edges as "seams", then selected the cross only and pressed U - "project from view", then selected all the circle, selected one face as "active" (deselect then re-select one face) and pressed U "follow active quad". Moved and scaled the UV islands as I wanted, because the textures are automatically tiled outside their UV space.



For the second part of the question, a common workflow is to set the size of the texture scaling the UV outside the boundaries, then when satisfied with the result, create a new UV map optimized for export and bake all texures in a new image with the new UV map, so you can optimize your texture space.

After you've arranged your UV map, create a new one (properties window, vertex tab, plus button in the UVMaps panel (I called it Target UV), select all and press U project from view. Create a material with an emission shader controlled by the floor image with the first UV, create a second image node contgrolled by the Target UV, load a blank image of the desired dimension, have it selected in the node editor (even if it's not connected to anything), switch to Cycles Render and press Bake button with emission option. This will create a new image texture with a complete UV Mapping inside the boundaries.



Plus, enable the "Node wrangler" add-on in the preferences, so that you can select an image node in the node editor and press Ctrl T to quickly create the UV and mapping nodes.